Find a quadratic polynomial whose zeroes are 2 - 5√2 and 2 + 5√2
Answers
Answer :
The required polynomial is x² - 4x - 46
Step-by-step explanation :
➤ Quadratic Polynomials :
✯ It is a polynomial of degree 2
✯ General form :
ax² + bx + c = 0
✯ Determinant, D = b² - 4ac
✯ Based on the value of Determinant, we can define the nature of roots.
D > 0 ; real and unequal roots
D = 0 ; real and equal roots
D < 0 ; no real roots i.e., imaginary
✯ Relationship between zeroes and coefficients :
✩ Sum of zeroes = -b/a
✩ Product of zeroes = c/a
________________________________
Given zeroes of the quadratic polynomial,
2 - 5√2 and 2 + 5√2
⇒ Sum of zeroes
= (2 - 5√2) + (2 + 5√2)
= 2 - 5√2 + 2 + 5√2
= 4
⇒ Product of zeroes
= (2 - 5√2) (2 + 5√2)
= 2(2 + 5√2) - 5√2(2 + 5√2)
= 4 + 10√2 - 10√2 - 5√2(5√2)
= 4 - 25(2)
= 4 - 50
= -46
The quadratic polynomial will be in the form of :
x² - (sum of zeroes)x + (product of zeroes)
x² - 4x + (-46)
x² - 4x - 46
The required polynomial is x² - 4x - 46